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According to the Electric Universe folk at the Thunderbolts web site, black holes do not exist. According to the EU folk, black holes are not even consistent with Einstein's general theory of relativity.

As intellectual cover for their position, the EU folk cite a series of papers that have been published by Stephen J Crothers.

It follows in similar fashion that expansion of the Universe and the Big Bang cosmology are inconsistent with General Relativity, as is easily demonstrated [24, 25].

Crothers is making a purely mathematical claim. He isn't talking about whether black holes actually exist or the Big Bang ever happened. He's saying the mathematical theory of general relativity predicts neither, and is inconsistent with both.

In this thread, we will consider that purely mathematical question. We will accept the general theory of relativity as a given, even though many physicists hope GR will ultimately be replaced by an even better theory that doesn't break down at the singularities. We will analyze the mathematical arguments put forth by Crothers and a few others, and we will identify some of the mathematical errors that Crothers has made. We will also construct mathematical counter-examples to some of his arguments.

According to Crothers, the truth about black holes is being suppressed:

Originally Posted by Stephen J Crothers

Neither the layman nor the specialist, in general, have any knowledge of the historical circumstances underlying the genesis of the idea of the Black Hole. Essentially, almost all and sundry simply take for granted the unsubstantiated allegations of some ostentatious minority of the relativists. Unfortunately, that minority has been rather careless with the truth and is quite averse to having its claims corrected, notwithstanding the documentary evidence on the historical record. Furthermore, not a few of that vainglorious and disingenuous coterie, particularly amongst those of some notoriety, attempt to dismiss the testimony of the literature with contempt, and even deliberate falsehoods, claiming that history is of no importance. The historical record clearly demonstrates that the Black Hole has been conjured up by combination of confusion, superstition and ineptitude, and is sustained by widespread suppression of facts, both physical and theoretical....

Although that's the sort of language we see more often at JREF than in a scientific journal, those were the opening words of the abstract for

That paper is very well-written, and its math is almost (but not quite!) correct.

That paper's central error is topological. I think I can explain that topological error to non-mathematicians. That explanation may also interest the physicists who know Abrams's conclusion was wrong but don't know exactly where he went wrong.

I didn't want to derail any of those threads with this mathematical side-show. The "Black holes" thread is about the mainstream theory of black holes, not the crackpot EU denial of black holes. The third thread listed is about the recent publication of EU pseudoscience in a particular journal; none of the papers I intend to discuss were published in that journal.

In this thread, I hope we can maintain our focus on the mathematics. To reduce redundancy, I will occasionally link to math that's already been posted in one of the above threads.

Intuitively speaking, topology is an abstraction of geometry that concentrates on the properties of geometric spaces that don't change when the space is subjected to certain kinds of continuous deformations.

A topological spaceWP consists of an arbitrary set X of points together with a topology (defined below) that, intuitively speaking, gives us a way of talking about which points are nearby. More formally, a topology T is a collection of subsets of X such that

the empty set is an element of T

X is an element of T

intersecting any finite number of elements of T gives you an element of T

intersecting any number of elements of T gives you an element of T

Example: Suppose X is the set of ordered pairs <x,y> where both x and y are real numbers. For any point p in X and for any positive real number ε, let N(p,ε) be the set of points whose Euclidean distance from p is strictly less than ε. (N(p,ε) is said to be the neighborhood of radius ε around p.) Suppose T consists of every set that can be obtained by taking the union of a (possibly infinite) set of such neighborhoods. Then X with topology T is the important topological space known as Euclidean 2-space or R2.

Example: Suppose X is the set of points on the surface of a globe. For any point p in X and for any positive real number ε, let N(p,ε) be the set of points whose great-circle distance from p is strictly less than ε. Suppose T consists of every set that can be obtained by taking the union of a (possibly infinite) set of such neighborhoods. Then X with topology T is another important topological space known as the 2-sphere.

If X is the set of points belonging to a topological space, and T is its topology, then the elements of T are said to be the open sets of that topological space.

If f is a mathematical function that maps one topological space into another, then f is said to be continuous if and only if: for every open set V of the second topological space, the set f-1V (consisting of the points in the first space that f maps into V) is an open set of the first space. (This definition of continuous functions is more general than the usual definition, but coincides with the usual definition on familiar metric spaces such as Euclidean 2-space and the 2-sphere.)

If the continuous function f is also a one-to-one correspondence (aka bijection) between the two topological spaces, and its inverse is continuous, then f is said to be a homeomorphism, and the two spaces are said to be homeomorphic to each other.

If two topological spaces are homeomorphic, then there exists a continuous function f that converts one of those spaces into the other, while its inverse function f-1 reverses that conversion. It is also possible to apply the inverse function first, and to reverse that conversion using f. That means the two homeomorphic spaces are topologically equivalent, and differ only with regard to the reversible stretching and squeezing that's performed by f.

Example. A doughnut (torus) is homeomorphic to a coffee cup, but is not homeomorphic to the 2-sphere or to Euclidean 2-space.

Example. The 2-sphere is homeomorphic to the surface of a cube, but is not homeomorphic to Euclidean 2-space.

Example. If you remove any single point from the 2-sphere, the topological subspace that remains is homeomorphic to Euclidean 2-space.

That last example is so important that I'm going to define an explicit homeomorphism for it. Let's think of the 2-sphere as a perfect sphere embedded in Euclidean 3-space, and let's think of the point we remove as the south pole. For any other point p, define r(p) as the great-circle distance from p to the north pole, define s(p) as the great-circle distance from p to the south pole, and define φ(p) as the longitude of p (with any prime meridian you like); if p is the north pole, then define φ(p) to be zero.

Define f(p)=<(r(p)/s(p)) cos φ(p), (r(p)/s(p)) sin φ(p)>. Then f(p) defines a one-to-one correspondence between the 2-sphere without its south pole and Euclidean 2-space. It's easy to check that f(p) is continuous and has a continuous inverse. Euclidean 2-space is therefore homeomorphic to the 2-sphere without its south pole.

Instead of omitting just the south pole, we could omit any closed disk that's centered on the south pole and define s(p) to be the great-circle distance from p to the circular boundary of that missing disk. With that modification, the definition of f(p) above would give us a homeomorphism between Euclidean 2-space and the 2-sphere without that closed disk.

Topologically speaking, it doesn't matter whether you remove a single point from the 2-sphere or an entire closed disk. Either way, you get a topological space that's equivalent to Euclidean 2-space.

In like manner, removing a single point from Euclidean 3-space gives you a topological space that's homeomorphic to what you get by removing an entire closed ball from Euclidean 3-space.

That italicized fact is the topological fact we need to explain where Abrams went wrong. Later on, we will also use the fact that the 2-sphere is not homeomorphic to Euclidean 2-space.

According to the Electric Universe folk at the Thunderbolts web site, black holes do not exist. According to the EU folk, black holes are not even consistent with Einstein's general theory of relativity.

__________________Almo!My Music Blog
"No society ever collapsed because the poor had too much." — LeftySergeant
"It may be that there is no body really at rest, to which the places and motions of others may be referred." –Issac Newton in the Principia

Crothers appears to be a rather argumentative sort of Australian who started getting a PhD in Physics at UNSW. Early on he got a bee in his bonnet about black holes, and it became an obsession such that he changed his topic to Relativity. Subsequently his iconoclastic ideas and his his crankiness about them, along with his rather unsweet personality caused the university to expel him and his thesis advisor to disown him. Since then he has been waging a war against the whole of physics and a goodly bit of mathematics, denouncing such as:

Originally Posted by crothers

I have also had some correspondence with a number of other sheepish relativists of no consequence. Not one offered any science, just the authority of Hawking, Penrose, Einstein, and others. All were rude, stupid and incompetent (precisely what they accused me of being, evidently taking umbrage for my return of the epithets). Other big-shots in black holes and big bangs, such as Thorne, Misner, Israel, Rees, Penrose, Hawking, Ellis, Wald, little-shots such as J. Moffat, J. Barrow, S. Carroll, R. d'Inverno, B. Shutz, some tiny-shots not worth a mention, and a few other scribblers of textbooks and popular science, simply ignore correspondence.

Every person he corresponded with is "inept" or mendacious.

He made a formal attempt to have the Australian government abort funding to the Australian International Gravitational Observatory. He had a short article to Sky and Telescope rejected; it is here, and forms a possible introduction to his work:

Ok, I'm with you so far on the homeomorphism. Interesting stuff; I always liked topology.

__________________Almo!My Music Blog
"No society ever collapsed because the poor had too much." — LeftySergeant
"It may be that there is no body really at rest, to which the places and motions of others may be referred." –Issac Newton in the Principia

If everyone you know is a jerk, it's you.Meaning Crothers, just in case it wasn't obvious.

__________________"As long as it is admitted that the law may be diverted from its true purpose -- that it may violate property instead of protecting it -- then everyone will want to participate in making the law, either to protect himself against plunder or to use it for plunder. Political questions will always be prejudicial, dominant, and all-absorbing. There will be fighting at the door of the Legislative Palace, and the struggle within will be no less furious." - Bastiat, The Law

Dr. Leonard S. Abrams was born in Chicago in 1924 and died on December 28, 2001, in Los Angeles at the age of 77. He received a B.S. in Mathematics from the California Institute of Technology and a Ph.D. in physics from the University of California at Los Angeles at the age of 45. He spent almost all of his career working in the private sector, although he taught at a variety of institutions including California State University at Dominguez Hills and at the University of Southern California. He was a pioneer in applying game theory to business problems and was an expert in noise theory, but his first love always was general relativity. His principle theoretical contributions focused on non-black hole solutions to Einstein’s equations and on the inextendability of the “Schwarzschild” solution. Dr. Abrams is survived by his wife and two children.

Abrams was wrong about the inextendability of the Schwarzschild solution, but he did some good stuff. In what follows, all quotations attributed to Abrams come from

The first paragraph of that paper's conclusion, with my editorial comments added in blue:

Originally Posted by Leonard S Abrams

We summarize the result of the preceding sections as follows. The K-F (Kruskal-Fronsdal) black hole is the result of a mathematically invalid assumption, explains nothing that is not equally well explained by SS(Schwarzschild spacetime), cannot be generated by any known process, and is physically unreal. Clearly, it is time to relegate it to the same museum that holds the phlogiston theory of heat, the flat earth, and other will-o’-the-wisps of physics.

That's quite a conclusion.

Although I'm not a physicist myself, it's my impression that many (most?) physicists think the "white-hole" half of Kruskal-Fronsdal spacetime is unlikely to be physically real. The controversial part of Abrams's conclusion is that he regards everything inside the gravitational radius (aka Schwarzschild radius, event horizon) as physically unreal and "the result of a mathematically invalid assumption."

It turns out that the mathematically invalid assumption was made by Abrams in his section 6. The first half of his paper is pretty interesting, however, and most of his Appendix A is okay as well. Before we get to his central error, I'll summarize the good stuff.

Several spacetime manifolds, all of them equivalent

In 1916, Karl Schwarzschild found the first nontrivial exact solution to Einstein's field equations, for a static spherically symmetric spacetime around an isolated, non-rotating, neutral star of mass m, regarded as a point mass. This is known as Schwarzschild's exterior solution. (He wrote a second paper that describes a solution for the interior of a star that isn't regarded as a point mass, but that interior solution isn't so relevant here.)

As is now well known, Schwarzschild's exterior solution suffers from a coordinate singularity at what is now called the Schwarzschild radius r=2m. That didn't seem terribly consequential at the time, because the Schwarzschild radius lay well inside all known stars. If you wanted to understand spacetime inside a star, you'd use the interior solution, which didn't have that particular coordinate singularity.

What's less well known is that, using the notational conventions I've been using in other threads, Schwarzschild's line-element (pseudo-metric) was:

That's an algebraically inconvenient way to write the pseudometric, so Abrams uses the following equivalent form, which he attributes to Brillouin:

Abrams defines that spacetime as SS, and refers to it as Schwarzschild's spacetime.

Transforming the r coordinate above by adding the Schwarzschild radius α yields the equation that most of us think of as the Schwarzschild metric:

Abrams refers to that as Flamm spacetime SF. Note well that, in Flamm spacetime, the origin of Schwarzschild spacetime has been transformed from r=0 to r=2m. According to Abrams, that means we must think of the entire 2-sphere at r=2m (and arbitrary t) as representing a single point, which he regards as the point mass.

Note also, however, that the 2-sphere at r=2m lies outside the Flamm spacetime, just as the point mass at r=0 lies outside the Schwarzschild spacetime. Abrams is attaching some extra-mathematical mental baggage to the mathematical manifolds he's defining.

Finally, Abrams defines Hilbert's spacetime as the spacetime that's mathematically identical to the Flamm spacetime, but with different mental baggage: Abrams accuses Hilbert of thinking the point mass lies at r=0, and explains how Abrams thinks Hilbert came to think that way. That's Hilbert's alleged error, as mentioned in the title of the paper.

As a mathematician, I have to ask: So what? Hilbert's mental states are no more relevant than Abrams's. What matters here are the manifolds (spacetimes) themselves. The Flamm and Hilbert spacetimes are obviously isometric. Both end at the Schwarzschild radius. Neither includes any points at the Schwarzschild radius, so their spatial slices are missing either a point (Flamm) or a closed ball (Hilbert) at their centers.

Does it matter whether the part that's missing is a point or a closed ball? Not at all: As noted at the end of my previous post, the topological space you get by removing a single point from Euclidean 3-space is homeomorphic to the space you get by removing a closed ball.

In his section 5, Abrams observes that Hilbert spacetime can be extended to a larger manifold that includes the Schwarzschild radius and points inside that radius. That extension bypasses the well-known coordinate singularity at the Schwarzschild radius of the Hilbert/Droste/Weyl metric (which is more popularly known as the Schwarzschild metric). Unfortunately, Abrams thinks the corresponding coordinate singularity at the central point mass of the original Schwarzschild spacetime is an irremovable "quasiregular singularity". Abrams therefore believes he has discovered an important difference between the Schwarzschild and Hilbert spacetimes.

In reality, the "quasiregular singularity" at the central point mass of the original Schwarzschild spacetime can be removed by allowing the radial coordinate r to go negative. You can understand why Abrams never considered that possibility, but he should have: Abrams calls the reader's attention to the fact that the radial coordinates of these spacetimes are not identical to the radial coordinates of Euclidean space. He should have realized that the usual assumptions we make about Euclidean coordinates may not apply.

If you want to see exactly how the original Schwarzschild spacetime can be extended by allowing r to go negative, you can read this recent paper that DeiRenDopa cited in another thread:

In the original version of Abrams's paper, section 6 begins as follows:

Originally Posted by Leonard S Abrams

If two space-times are to be equivalent, it is certainly necessary that they be isometric i.e., that there exists a diffeomorphism from one to the other that carries the metric of one into the metric of the other. And since the presence of singularities of the manifold geometry is unaffected by diffeomorphisms, it is also necessary that equivalent space-times have the same “singularity structure”, i.e., the same singularities as one approaches corresponding boundary points. Now, SS and SH are isometric under Tα,

So far, so good. All of that is true. Note, however, that the mathematical definition of a spacetime manifold does not mention any explicit "singularity structure". The singularities and their structure must be inferred from the manifolds themselves. Since SS and SH are mathematically equivalent as manifolds, we must infer the same "singularity structure" for both of them.

Originally Posted by Leonard S Abrams

but as shown in the preceding section, SH has no singularity corresponding to the quasiregular singularity at r = 0 in SS. Consequently, SS and SH are inequivalent. Since it was shown in Sect. 2 that SS is the space-time of a point mass, it follows that SH and its analytic extension (SK−F ) are not.

That's where Abrams goes off the rails. He's comparing the extensibility of SH (which is a mathematical fact) to the mental baggage he's attached to the Flamm and Schwarzschild spacetimes, which is a mistake. Abrams thinks the Flamm and Schwarzschild spacetimes are inextensible, but he never actually proved that, and he's wrong.

Abrams must have realized there was something wrong with his section 6, because he published an erratum that replaces his entire section 6 with a paragraph that starts like this:

Originally Posted by Leonard S Abrams

By inspection, SS and SH are isometric via Tα and thus equivalent.

Abrams should have stopped right there, and drawn the obvious conclusion.

But he went on:

Originally Posted by Leonard S Abrams

However, it was shown above that due to the difference in the topology of their boundaries, they are associated with different singularity structures. Thus, the universes (US and UH) corresponding to SS and SH (with their indicated boundaries) are inequivalent...

That's just wrong.

The boundaries aren't part of the spacetime manifolds, so Abrams is discussing his mental baggage. Mathematically, it is more correct to imagine all of those boundaries as 2-spheres (for any t) than to imagine any of those boundaries as points, because all of those manifolds can be extended by attaching a 2-sphere (and points within). None of those manifolds can be extended by attaching a single point.

The "singularity structures" aren't part of the spacetime manifolds either. Singularities must be inferred from the spacetime manifolds. Because these spacetime manifolds are equivalent, they have the same singularities. Full stop. Any differences that Abrams may think he perceives come from the mental baggage he's attached to these manifolds.

In my next post, I'll point out some other assumptions that Abrams should not have made. We'll get to the Crothers papers eventually, which is where the real fun begins.

intersecting any finite number of elements of T gives you an element of T

intersecting any number of elements of T gives you an element of T

Please replace the highlighted word with "Taking the union of".

We're almost done with the hard stuff. Tomorrow I'll outline an explicit definition of a manifold that extends Flamm spacetime (which is better known as Schwarzschild spacetime) to include the Schwarzschild radius (event horizon) and part of its interior, and then I'll start on the fun stuff: identifying some of Crothers's more obvious mistakes.

In this post, we'll look at the assumptions and constraints that led to what is now known as Schwarzschild's exterior solution of Einstein's field equations for spacetime around a non-rotating, electrically neutral star or black hole.

In the early years, no one gave any thought to the possibility of a black hole. That changed in 1931, when Chadrasekhar realized that large white dwarfs would eventually collapse under their own gravity. In 1939, Oppenheimer predicted that large neutron stars would collapse into black holes. Unfortunately, Oppenheimer misinterpreted the Schwarzschild coordinate singularity to mean time stops at the Schwarzschild radius, and that misinterpretation is still being promoted by the electric universe folk and some others.

Consider now the particular U consisting of a single uncharged, nonrotating, nonradiating point mass (whose Newtonian gravitational mass will henceforth be denoted by ‘m’). Historically, the conditions regarded as distinguishing the space-time (MU , gU ) of this U from those of all others where originally formulated by Einstein [8], and together with those implicit in that formulation were enumerated by Finkelstein [9].

Abrams assumed the following conditions:

The spacetime is static.

The spatial part of spacetime is spherically symmetric.

The (pseudo-) metric has Lorentz signature.

One of the coordinates represents global time.

The metric satisfies Einstein's field equations for empty space.

The metric is asymptotically flat (at infinity).

The metric coefficients are analytic functions of the spatial coordinates.

The coefficient for the dΩ2 term is greater than or equal to (2m)2 for all positive values of the radial coordinate.

A single set of coordinates covers all of spacetime.

The first 6 conditions (and possibly the 7th) were assumed by Schwarzschild. Abrams assumed condition 8 in his equation (11); it is not implied by the first 7 conditions. Condition 9 was implicit.

The assumption of static spacetime

I got the definition of static spacetime wrong in another thread, so let me try again. For the purposes of this thread, I think it's enough to say that a static spacetime means none of the metric coefficients depend on time, and dt appears only within the dt2 term (so there are no cross terms involving dt dr, for example).

In 1915, when Einstein formulated this problem, he could not have known that a static solution actually exists. Fortunately, Schwarzschild's exterior solution was static, provided the star's radius was greater than 2m. That was true for all known stars.

(If you're willing to contemplate a weird interchange of time and radial coordinates inside the Schwarzschild radius, then you can say that Schwarzschild's solution extends inside the Schwarzschild radius, but that interchange of coordinates makes the solution non-static inside the Schwarzschild radius, and you still have to deal with the coordinate singularity at the Schwarzschild radius. To keep things simple, I'm not going to go there.)

Contrary to Abrams's central claim, Schwarzschild's spacetime manifold can be extended to include the event horizon and spatial points inside that horizon, but that extension is non-static inside the event horizon. To understand spacetime in the near vicinity of a black hole, we have to abandon the assumption that spacetime is static.

Many of the electric universe folk appear to be motivated in part by hope that cosmology is static or steady-state or at least stationary (so it repeats itself). As Michael Mozina has demonstrated, some electric universe folk have accused mainstream cosmologists of being creationists, just because mainstream cosmology no longer assumes spacetime is static.

Lemaître was not a creationist, but he was a Catholic priest. It is entirely possible that Lemaître's religious beliefs allowed him to think outside the static box that prevented so many of his contemporaries from finding non-static solutions to Einstein's field equations. So what? Lemaître's religious beliefs have nothing to do with mathematical facts.

It's a mathematical fact that, according to Einstein's field equations, the spacetime manifold surrounding a point mass is non-static. It's a mathematical fact that the only way to regard that spacetime manifold as static is to cut out the part that surrounds the point mass, all the way out to and including the Schwarzschild radius. It's also a mathematical fact that all attempts to justify that surgery on mathematical grounds have failed.

In particular: The attempts by Abrams and by Crothers do not stand up under mathematical scrutiny.

Their failure to prove that the spacetime manifold must end at the Schwarzschild radius does not automatically imply the existence of a spacetime manifold that includes the Schwarzschild radius and its interior. Our next step is to give a mathematical proof of that spacetime manifold's existence.

Mathematical proof of existence

Spacetime manifolds are mathematical objects. We can prove their mathematical existence follows from the laws of logic and the axioms of mathematics, but we cannot use mathematics alone to prove a spacetime manifold accurately describes the physical universe. Whether something exists in a physical sense is a question for science, not mathematics.

On the other hand, science tells us the laws of general relativity do a pretty good job of describing the physical universe as we know it. That's why the mathematical existence of a spacetime manifold should be taken more seriously than electric universe pseudomath and pseudoscience.

As noted in my original post, many electric universe folk have been claiming that black holes are incompatible with the theory of general relativity. The most straightforward way to refute that claim is to present mathematical proof that a certain spacetime manifold exists and has all of the properties that define a black hole.

I'll outline and then sketch that proof here. I'll leave some of the details as exercises, mainly because inclusion of all details would lengthen the proof and make it harder to read. If you're capable of understanding the details I omit, then you are probably capable of filling them in on your own, and many of you have already done homework of that sort. I'll give hints in blue.

Outline of proof

Define a differentiable manifold M.

Define the points of M.

Define a topology for M.

Prove the properties required of a topological manifold:

Hausdorff

paracompact

locally Euclidean

Define a set of charts (coordinate systems) that cover M.

Prove those charts comprise an atlas for M.

Define a spacetime manifold M.

Define a Lorentzian pseudo-metric on M.

Prove the metric satisfies Einstein's field equations for empty space.

Prove M has the properties that define a black hole.

Definition of M as a differentiable manifold

It's convenient to have a distinct name for every point of the manifold we're defining. There are too many points to list each name individually, so we'll use a systematic naming process based on coordinates. The points of the manifold can be arbitrary mathematical objects, so we'll identify each point with its name (which will be an ordered list of 4 coordinates).

Let X0 be the set of all 4-tuples of the form <τ,r,θ,φ> where τ, r, θ, and φ are real numbers in the range

-∞ < τ < ∞
0 < r < ∞
0 < θ < π
-π < φ < π

and π is the Greek letter pi (representing the ratio between a Euclidean circle and its diameter). Let X1 be the set of all <τ,r,θ,π> where τ, r, and θ are real numbers in the range

-∞ < τ < ∞
0 < r < ∞
0 < θ < π

Let X2 be the set of all <τ,r,0,0> where τ and r are real numbers in the range

-∞ < τ < ∞
0 < r < ∞

and let X3 be the set of all <τ,r,π,0> where τ and r are as above. Define

X = X0 ∪ X1 ∪ X2 ∪ X3

The existence of those sets is an immediate consequence of the usual axioms for set theory and mathematics. Except where noted, the existence of everything we define will be an obvious consequence of basic axioms.

X0 will be the domain of our primary chart. We'll need a second chart whose domain is an open set that covers the meridian points in X1 and the polar points in X2 ∪ X3. Let the domain of that second chart be obtained by rotating the domain of the first chart as in exercise 7 of the exercises I formulated for Farsight. The domains of the two charts will then be connected by an obvious homeomorphism.

For every point p=<τ,r,θ,φ> in X0 and for every positive real number ε, define the open neighborhood N(p,ε) to be the set of all points q=<τ',r',θ',φ'> in X0 such that

These are square neighborhoods. The round neighborhoods you might have been expecting can be obtained by taking an infinite union of these square neighborhoods. Similarly, square neighborhoods can be obtained by taking an infinite union of round neighborhoods.

We also need an analogous definition of open neighborhoods for points on the meridians and at the poles, but I'll leave that as an exercise for readers.

Points on the meridians and poles add a lot of detail without adding much insight, so I'll basically ignore X1 ∪ X2 ∪ X3 from now on.

Define the open sets of M as the set of all (finite and infinite) unions of open neighborhoods, together with the empty set (which can be regarded as a degenerate union of zero open neighborhoods).

Exercise: Prove the intersection of any two open sets is an open set.

The union of all open neighborhoods is X, so we have defined a topology on X. We must now prove that the topology is Hausdorff, paracompact, and locally Euclidean.

Exercise: Prove the topology is Hausdorff. (You have to show that for every pair of distinct points p and q, there's an open neighborhood U around p and an open neighborhood V around q such that U and V don't overlap.)

Exercise: Prove the topology on X is consistent with the usual Euclidean distance metric on X.

Exercise: Prove the topology on X is paracompact. (Hint: Every metrizable space is paracompact, and the previous exercise says the topological space we've defined is metrizable, so this is just modus ponens. You don't even have to know what paracompactness means!)

Let f: X0 → R4 be the function that maps <τ,r,θ,φ> to <τ,r,θ,φ>.

Exercise: Prove that if V is an open subset of R4, then the set of points that f maps into V is an open subset of X.

Exercise: Prove that if U is an open subset of X0, then f(U) is an open subset of R4

Exercise: Prove that f defines a one-to-one correspondence between X0 and f(X0).

Exercise: Prove that f is a homeomorphism between X0 and f(X0). (Hint: That's just a succinct summary of the three exercises above!)

Exercise: Define an analogous function g for the subset of X obtained by rotating X0 as in exercise 7. (Hint: Define g as the composition of f with that rotation's inverse.)

The exercises above prove that the topological space we have defined is locally Euclidean. They also provide us with two homeomorphisms f and g whose combined domains cover all of X.

Exercise: Let U be the intersection of X0 with the domain of g, and define h: f(U) → g(U) by h(x) = g(f-1(x)). Prove that h is a one-to-one correspondence.

Exercise: Prove that h is differentiable.

Repeat both of the above exercises for h: g(U) → f(U) defined by h(x) = f(g-1(x)).

The exercises above prove that f and g have the composition property required of charts. Taken together, the domains of f and g cover all of X.

Define M to be the differentiable manifold whose set of points is X, whose topology is as defined above, and whose atlas consists of the two charts f and g.

Definition of a Lorentzian manifold

We have now defined a Riemannian (locally Euclidean) manifold M. What we really wanted, however, was a pseudo-Riemannian (locally Lorentzian) manifold M. Instead of repeating the above definition with the Minkowski pseudo-metric on R4 substituted for the Euclidean metric throughout, I'm going to leave that as an (uninteresting) exercise for the reader. Almost all of that definition is exactly the same as for a locally Euclidean manifold, and the few steps that aren't exactly the same are almost exactly the same.

The more important fact is that we have not yet defined a metric (or pseudo-metric) on the manifold.

Definition of M as a spacetime manifold

To define a Lorentzian pseudo-metric tensor field on M, it suffices to define the tensor field for both the domain of the chart f (which is X0) and for the domain of the chart g. With f and g the same up to a spatial rotation, it's enough to define the tensor field on X0.

Define the Lorentzian pseudo-metric tensor field on X0 by

where m is a non-negative parameter to be discussed later and

The definition above assigns a pseudo-metric tensor to every point of M. The nonzero components of that tensor are

To prove that g is a pseudo-metric tensor, we must prove it's bilinear, symmetric, and nondegenerate.

Bilinearity is obvious because we've written it as a 4x4 matrix with only 6 nonzero elements. Symmetry is obvious because the matrix is symmetric.

Exercise: Prove g is nondegenerate. (Hint: Calculate the determinant of the matrix of coefficients, and show that it's nonzero everywhere in X0.)

The pseudo-metric defined above is not static, because its coefficient for dτ dr is nonzero.

The next step is to prove g satisfies Einstein's field equations for empty space:

That's just a calculation, but it's a chore to do the straightforward way (by using g to calculate 40 Christoffel symbols, using those to calculate 20 components of the Riemann tensor, and using those to calculate the Ricci tensor Rμν and the Ricci scalar R).

I'm going to cheat. In his paper cited earlier, Abrams proves that all metrics of a certain form satisfy the vacuum field equations. Similar proofs are found in standard references such as Misner/Thorne/Wheeler (section 23.2 and Box 23.3) or Wald (section 6.1).

I'm not going to cheat very much, though. Exercise 25, in the "Black holes" thread, proves that the Painlevé-Gullstrand metric defined above is equal to the Schwarzschild metric outside the event horizon, and everyone seems to agree that the Schwarzschild metric satisfies Einstein's equations.

That proof also reveals that the r, θ, and φ coordinates of our spacetime M are identical to the corresponding Schwarzschild coordinates.

Exercise 26, in the "Black holes" thread, proves that the Schwarzschild manifold (when restricted to the part outside the Schwarzschild radius) is a proper submanifold of the spacetime manifold we've defined above.

That completes our proof that the Schwarzschild manifold is extensible, contrary to the central claim made by Abrams and by Crothers.

Proof that M has the properties of a black hole

The free parameter m represents the total mass that's present at the center of each spatial slice of M. That can be confirmed by examining the asymptotically Newtonian gravitational field implied by the manifold M, or we can cheat again by using the meaning of m in the Schwarzschild metric, which is equivalent to the one we defined wherever both are defined.

Let's calculate the coordinate speed of radial light at the event horizon of our manifold. (That coordinate speed of light is coordinate-dependent, but we already know that the r coordinate of our coordinate system is identical to the r of Schwarzschild coordinates, and we also know that a positive dτ represents a positive (future-directed) increment of time.) Light travels along null geodesics, so ds=0 for light. For radial light, dθ=dφ=0 as well. At the event horizon, β = 1. Hence

At the event horizon, therefore, dr=0 (for outgoing light) or dr=-2 dτ (for ingoing light). The coordinate speed of outgoing light is dr/dτ=0, and the coordinate speed of ingoing light is dr/dτ=-2.

Outgoing light isn't making any outward progress at all. Looks like a black hole to me.

It is plainly evident, res ipsa locquitur, that the standard claims for black holes and Big Bang cosmology are not consistent with elementary differential geometry and are consequently inconsistent with General Relativity.

That sounds intriguing. I'm always open to learning more differential geometry. If it demolishes the standard theory of black holes and big bang cosmology, so much the better. Breaking things can be fun.

The following section, whose title is "Spherical symmetry of three-dimensional metrics", begins with this sentence:

Originally Posted by Stephen J Crothers

Denote ordinary Efcleethean∗ 3-space by E3.

Efcleethean?

Crothers provided this helpful footnote:

Originally Posted by Stephen J Crothers

∗For the geometry due to Efcleethees, usually and abominably rendered as Euclid.

If it is possible to have M3 and E3 in one-to-one correspondence, as you claim, then please provide us with such a mapping. In fact I'll make it even less onerous - simply provide us with a one-to-one mapping between the 1D circle and 1D Euclidean space.

The mapping you request is developed in the paper based upon the hypothesis of a one-to-one correspondence between E3 and M3, and associated rigid rotations in one corresponding to rigid motions in the other so that geodesics are mapped into geodesics.

So that was all much ado 'bout nuthin'. The Crothers paper deals only with Euclidean 3-manifolds that are homeomorphic to E3, and my contrary impression came about only because the paper is poorly written. I guess I'll just have to put up with the poor writing, because I really do hope to learn how the standard theory of big bang cosmology is incompatible with elementary differential geometry.

I won't keep you in suspense: He doesn't. The paper fails to deliver on the promises made in its third paragraph. Most of the paper is devoted to Schwarzschild spacetime, and most of that is just a restatement of what Crothers wrote in his very first paper, which was itself mostly a rehash of the Abrams paper I went over in post #8 and post #11. Crothers repeats the errors in the Abrams paper and adds a few more of his own.

Here's the final sentence of section 8 ("That the manifold is inextendable"):

Originally Posted by Stephen J Crothers

Thus, the Schwarzschild manifold described by (20) with (22) (and hence (8)) is inextendable.

Equation (8) is the familiar metric for the Schwarzschild manifold. In post #12 above, I sketched a fairly detailed definition of a spacetime manifold that extends that Schwarzschild manifold.

When Crothers tells me the thing I've done can't be done, I begin to doubt his unsupported claims.

I can't honestly say I've learned nothing about differential geometry from Crothers. Working through his calculations and identifying his errors had some educational value, but the process has been less educational than entertaining.

Clinger, I am not doing well following your argument. Unfortunately, my fading mathematics skills are not up to the task but I am getting the drift of your conclusions. Thank you.
By the way, Efcleethees as a transliteration of Ευκλείδης is pathetic in several ways. First, rendering Ευ as Ef is quite bizarre since in ancient Greek the υ here is sounded as something like oo in English (the f sound came into Greek during the middle ages). Similarly, δ as th is something that comes from Greek of the middle ages; ancient Greek sounded δ as d. Using c instead of k comes about because of our inheriting Euclid's name through Latin (this has happened to many words, like the prefix cardio_ = Greek kardio_). The use of ee has absolutely no basis whatsoever. If one were to be a stickler regarding ther transliteration of ancient Greek words, Ευκλείδης would be rendered as Eukleides. His scholarship seems to be as careless and lacking in this area as it is in his analysis of black holes.

That paper contains four technical sections, plus a dedication to Leonard S Abrams and an epilogue that explains how Crothers became interested in general relativity through the Adams paper I discussed previously. The first two technical sections basically repeat what Adams wrote, adding only some snark and a sloppy reformulation (that I believe to be incorrect) of one relatively unimportant formula.

In section 3, Crothers strikes out on his own. Here's what remains of the first half of section 3 after snark, redundancy, and irrelevancy have been excised:

Originally Posted by Stephen J Crothers

3 Implications for gravitational collapse

As is well known the gravitational potential Φ for an arbitrary metric is

from which it is concluded that gravitational collapse occurs at Φ = 1.

...snip...

From (21),

so in the weak far field,

and so

...snip...

According to (23), at r=r0, g00 = 0 and Φ = ½. The weak field potential approaches a finite maximum of ½ (i.e. ½c2), in contrast to Newton's potential. The conventional concept of gravitational collapse at rs = α is therefore meaningless.

I think Crothers got Φ = ½ by using his "weak far field" approximation under the assumption that C(r0)=(2m)2, but that weak field approximation is obviously the wrong formula to use for calculations in the strong field at r=r0. (Crothers takes that value of the radial coordinate r to be the location of the point mass, when it's actually the event horizon, but that doesn't matter here: it's a strong field either way).

The correct calculation of Φ = 1 appears to refute the main point Crothers is trying to make here. (I'm not entirely sure what point Crothers is trying to make here. Perhaps a physicist could explain it to me. Crothers couldn't.)

Finally, g00 is coordinate-dependent. (That's an observation, not a criticism, because the gravitational potential has to be coordinate-dependent.) Crothers thinks he's stuffed all of that coordinate-dependence into C(r), but that's true only for coordinate systems that satisfy a number of assumptions. Crothers made some of those assumptions explicit, but quite a few others are implicit. One of those assumptions is that the coordinates take advantage of spherical symmetry so he can ignore all but the time and radial coordinates; that's fine. Another assumption is that the metric is static. That rules out all coordinate systems that cover the event horizon and its spatial interior.

So I don't even believe equation (21) can be true in general.

The second half of section 3 is constructed atop that pile of fail. Some excerpts:

Originally Posted by Stephen J Crothers

Time stops at the Schwarzschild point-mass. The conventional concepts of the Schwarzschild sphere and its interior are meaningless.

...snip...

Applying (31) to the Kruskal-Szekeres extension gives rise to the absurdity of an infinite acceleration at r = α where it is conventionally claimed that there is no matter and no singularity. It is plainly evident that gravitational collapse terminates at a Schwarzschild simple point-mass, not in a black hole.

...snip...

The correct conclusion is that gravitational collapse terminates at the point-mass without the formation of a black hole in all general relativistic circumstances.

That's an extraordinary claim, because it contradicts all mainstream research into the implications of general relativity for black holes. When I can identify obvious errors in the chain of reasoning that's alleged to support the claim, I'm inclined to regard the extraordinary claim as unproven.

When I've done the math myself, and found that correct mathematics refutes most of what Crothers has to say, I'm inclined to regard the extraordinary claim as incorrect.

When I survey a fair sample of the 17 papers Crothers has published in Progress in Physics, and look at the magnificent effort Crothers has put into resisting correction by knowledgeable relativists and mathematicians, I'm inclined to regard Crothers as a crackpot.

Black holes are now “seen” everywhere by the astronomers, even though no one has ever found an event horizon anywhere. Consequently, public opinion has been persuaded that the black hole is a fact of Nature and that anyone who questions the contention must be a crackpot. It has become a rather lucrative business, this black hole. Quite a few have made fame and fortune peddling the shady story.

Yet it must not be forgotten that all the arguments for the black hole are theoretical, based solely upon the erroneous Hilbert solution and the meaningless Kruskal-Szekeres extension on it. One is therefore lead to wonder what it is that astronomers actually “see” when they claim that they have found yet another black hole here or there.

At the event horizon, therefore, dr=0 (for outgoing light) or dr=-2 dτ (for ingoing light). The coordinate speed of outgoing light is dr/dτ=0, and the coordinate speed of ingoing light is dr/dτ=-2.

Outgoing light isn't making any outward progress at all. Looks like a black hole to me.

Since I am unable to follow the mathematical argument here, please indulge me to ask a question in (non-mathematical) layman's terms.
There are many well established equations in physics that have been confirmed experimentally. Of course, this includes GR. However many of these equations are known to break down under extreme conditions. A naive example of this would be pV = k, which is valid for a wide range of pressures but breaks down at extreme pressures. When GR is applied at the extreme conditions of a black hole the mathematics may work, but is it not true that we have no way of knowing if it is actually valid under these extreme conditions?

Since I am unable to follow the mathematical argument here, please indulge me to ask a question in (non-mathematical) layman's terms.
There are many well established equations in physics that have been confirmed experimentally. Of course, this includes GR. However many of these equations are known to break down under extreme conditions. A naive example of this would be pV = k, which is valid for a wide range of pressures but breaks down at extreme pressures. When GR is applied at the extreme conditions of a black hole the mathematics may work, but is it not true that we have no way of knowing if it is actually valid under these extreme conditions?

That is correct, we cannot yet know that with certainty. We have no reason to think it breaks down at the event horizon (though we do have reason to think it will break down at the Planck scale), but without experimental tests, we cannot be sure. But that's what it would take to prevent a black hole: for GR to be wrong. And if it is wrong, we have nothing to replace it with right now. Even if we did, it would need to agree with GR where we can test it, and we would still have no way of choosing between GR and such an alternative without testing those extreme conditions, which we can't yet do.

__________________"As long as it is admitted that the law may be diverted from its true purpose -- that it may violate property instead of protecting it -- then everyone will want to participate in making the law, either to protect himself against plunder or to use it for plunder. Political questions will always be prejudicial, dominant, and all-absorbing. There will be fighting at the door of the Legislative Palace, and the struggle within will be no less furious." - Bastiat, The Law

That is correct, we cannot yet know that with certainty. We have no reason to think it breaks down at the event horizon (though we do have reason to think it will break down at the Planck scale), but without experimental tests, we cannot be sure. But that's what it would take to prevent a black hole: for GR to be wrong. And if it is wrong, we have nothing to replace it with right now. Even if we did, it would need to agree with GR where we can test it, and we would still have no way of choosing between GR and such an alternative without testing those extreme conditions, which we can't yet do.

Is there any possibility that the Large Hadron Collider might shed some light on this question?

Is there any possibility that the Large Hadron Collider might shed some light on this question?

Unless something very surprising happens, I don't think so.

__________________"As long as it is admitted that the law may be diverted from its true purpose -- that it may violate property instead of protecting it -- then everyone will want to participate in making the law, either to protect himself against plunder or to use it for plunder. Political questions will always be prejudicial, dominant, and all-absorbing. There will be fighting at the door of the Legislative Palace, and the struggle within will be no less furious." - Bastiat, The Law

At the event horizon, therefore, dr=0 (for outgoing light) or dr=-2 dτ (for ingoing light). The coordinate speed of outgoing light is dr/dτ=0, and the coordinate speed of ingoing light is dr/dτ=-2.

Outgoing light isn't making any outward progress at all. Looks like a black hole to me.

Since I am unable to follow the mathematical argument here, please indulge me to ask a question in (non-mathematical) layman's terms.

Ziggurat answered your non-mathematical (scientific) question.

The trouble you're having with the mathematical argument is likely to be my fault.

The steps of that argument go as follows:

The Painlevé-Gullstrand metric is equal to the familiar Schwarzschild metric outside the event horizon.

That's exercise 25 in the "Black holes" thread. Given exercise 24, the proof of exercise 25 is just a trivial calculation using high school algebra.

The r, θ, and φ coordinates of the Painlevé-Gullstrand metric are identical to the r, θ, and φ coordinates of the Schwarzschild metric (by exercise 25).

The Painlevé-Gullstrand metric satisfies Einstein's field equations for empty space.

(I didn't prove that, because the proof is both tedious and uncontroversial. Even Crothers agrees that the familiar Schwarzschild metric satisfies Einstein's field equations for empty space. Combined with exercise 25, that implies that the Painlevé-Gullstrand metric satisfies Einstein's field equations outside the event horizon. The coefficients of the metric are analytic in the coordinates, so the Painlevé-Gullstrand metric must satisfy the field equations at the event horizon as well. That's all we need for the following steps.)

According to general relativity, light follows null geodesics.

For null geodesics, ds=0.

For radial geodesics, which are directed entirely toward or entirely away from the center of spherical symmetry, the conventional spherical coordinates θ (elevation) and φ (azimuth) don't change.

For radial geodesics, therefore, dθ=dφ=0.

At the event horizon, r=2m so β=1.

Substituting ds=dθ=dφ=0 and β=1 into the Painlevé-Gullstrand metric, we find that null radial geodesics at the event horizon satisfy

That means one of the two factors must be zero.

That means dr=0 or dr=-2dτ.

That means dr/dτ=0 or dr/dτ= - 2.

That means the world line for inwardly directed radial light has coordinate velocity dr/dτ= - 2 at the event horizon. (The radial coordinate for inwardly directly radial light at the event horizon is decreasing as the time coordinate τ advances into the future.)

That means the world line for outwardly directed light has coordinate velocity dr/dτ=0 at the event horizon. (The radial coordinate for outwardly directly radial light at the event horizon isn't changing.)

That means a photon that's emitted radially outward at the event horizon never escapes beyond the event horizon. It stays forever at the event horizon.

That also means a photon that's emitted radially inward at the event horizon proceeds rapidly toward the central singularity, contra Crothers and Farsight.

If you don't understand how one of those steps follows from preceding steps, or you don't understand how one of those steps is derived, please let me know which steps require elaboration.

I am glad that S. Crothers is being discussed here. I have corresponded with him a bit on email and read over his whole website pretty thoroughly. I have to admit that I agree with him that black-holes should not be considered a good prediction of General Relativity, but I have different reasons then he does for thinking this.

I would love to discuss in another thread perhaps why I think this is so, but for now it seems more worthwhile to go over what I make out to be the main claims of Crothers and why I either agree or disagree with those claims.

1. He challenges the legitimacy of the stress-energy pseudo-tensor.

I think his analysis is in error on this point as he sets up various straw-man arguments. This is kind of technical so I will leave out the messy details unless anyone is interested about discussing it.

2. He challenges the legitimacy of using multiple bodies in GR as there are no known solutions or existence theorems on such solutions.

This is very bad logic. The problem is he is not thinking like a physicist here. A test mass in motion in a gravitational field is very well described in GR using the geodesic equation. Two masses and good predictions, no big deal.

His complaint about using the Schw. Sol. for the geometry exterior to two masses in orbit about each other is also off because of his peculiar aversion to using approximations when talking about physics (which is very annoying btw).

3. He claims that the r in the Schwarzschild solution is the curvature of radius.

Agreed. Calculated it myself and it is. How much emphasis you put into the significance of this fact is up for debate I suppose though.

4. Complaint about Ric=0. His complaint is that since there is no mass or energy where Ric=0 is solved, how can one then find a solution to such a situation that includes mass.

This kind of gets into territory of why I think black-holes are not a good physics prediction of GR, so let me shy away from that for now and just state the following.

It has been proven very conclusively that exterior to a spherically symmetric mass all solutions of the metric are equivalent to Schw.'s sol. What is the name of that damn theorem? Oh well, forgot.

My point is, assertions about the stress-energy tensor being zero in one region and yet having curvature in that same region being a problem are false. There is no problem because the source for the field would then be in another region giving the curvature. Let me stop there.

5. He challenges whether Astronomers have detected BH's or not because the test of a BH behaving as per current understanding is not whether there is a certain amount of energy-density in a given region (that does not test the predictions about BH's as per current consensus understanding), but tests of things like whether there is an event horizon or singularity (or singularities perhaps more realistically speaking?). Further, it is a complaint of his that both the event horizon and any singularities are unfalsifiable.

I agree with the complaint that just because a region has such and such a density of energy does not show that there is a BH there acting in the way that BH's are supposed to act according to current consensus. The problem is that energy density in some region is not the same as observing a singularity or event horizon (or perhaps other putative aspect of BH's). This is true.

I do agree that singularities are unfalsifiable, but I think that event horizons in principle constitute a falsifiable prediction. The aspect of light not being able to escape would probably be unfalsifiable, but it is asymptotically falsifiable to show that closer and closer to such a horizon things get smeared out as it were.

I only hope that I have chosen well his complaints and adequately restated his arguments. Good Luck!

__________________I learned much from the Order of the Jesuits. Until now, there has never been anything more grandiose, on the earth, than the hierarchical organization of the Catholic church. I transferred much of this organization into my own party.

Oh, one note, the unfalsifiability of either the event horizon of singularity also has a practical side to it. Sure, you could go into a putative BH's horizon, but since you would not make it out (unless you think the wormhole models are correct), you would not be able to tell anyone, unless I guess there were no singularity or event-horizon, hmm, does that mean they are falsifiable after all?

__________________I learned much from the Order of the Jesuits. Until now, there has never been anything more grandiose, on the earth, than the hierarchical organization of the Catholic church. I transferred much of this organization into my own party.

I would love to discuss in another thread perhaps why I think this is so, but for now it seems more worthwhile to go over what I make out to be the main claims of Crothers and why I either agree or disagree with those claims.

Thank you. I have something to add to your third note.

Originally Posted by tensordyne

3. He claims that the r in the Schwarzschild solution is the curvature of radius.

Agreed. Calculated it myself and it is. How much emphasis you put into the significance of this fact is up for debate I suppose though.

The radial coordinate r is defined in terms of the area of a 2-sphere whose center isthat's symmetric around the central singularity; it is not defined as the Euclidean distance from that singularity to the 2-sphere.

That fact is acknowledged by standard textbooks. In Misner/Thorne/Wheeler, r is defined by equation (23.9'). In Wald, r is defined by equation (6.1.3). As Wald wrote:

Originally Posted by Robert M Wald

In flat, three-dimensional Euclidean space, r would also be the value of the radius of the sphere, i.e., the distance from the surface of the sphere to its center. However, in a curved space, a sphere need not have a center...and even if it does, r need not bear any relation to the distance to the center. Nevertheless, we shall refer to r as the "radial coordinate" of the sphere.

MTW's and Wald's textbooks were published long before Crothers published his first paper. When Crothers writes things like

Originally Posted by Stephen J Crothers

None of the relativists have understood this, including Einstein himself.

he is either ignorant or lying. Crothers has written things like that in just about every paper he's published. (The quotation above comes from "A brief history of black holes", which I cited in the OP and again in post #18.)

Hmm, I wonder though, do you have any references showing prior understanding by the physics community at large that the r in the Schw. Sol. is the Gaussian Curvature of radius (assuming you agree with that determination, but it is easy to check)?

I find that the r in the S. Sol. being the curvature of radius an interesting notion myself. It tells you how spherical trigonometry should act for instance.

Does one define the r as being the Aerial Coordinate, or any of its brethren, or as the Curvature of radius of the two-sphere about the central point. Both definitions of r will give the same answer for Schwarzschild Geometry, the interesting thing about interpreting r as radius of curvature is that it only works for spherically symmetric geometries.

__________________I learned much from the Order of the Jesuits. Until now, there has never been anything more grandiose, on the earth, than the hierarchical organization of the Catholic church. I transferred much of this organization into my own party.

__________________I learned much from the Order of the Jesuits. Until now, there has never been anything more grandiose, on the earth, than the hierarchical organization of the Catholic church. I transferred much of this organization into my own party.

The object is SgrA*, as in a point source within the SgrA complex, which is the "A" (radio) source in the constellation of Sagittarius. "Sag" is used, sometimes, as an abbreviation for Sagittarius, in the name of (other) astronomical objects, e.g. the Sagittarius Dwarf Elliptical Galaxy (Sag DEG).

Hmm, I wonder though, do you have any references showing prior understanding by the physics community at large that the r in the Schw. Sol. is the Gaussian Curvature of radius (assuming you agree with that determination, but it is easy to check)?

I find that the r in the S. Sol. being the curvature of radius an interesting notion myself. It tells you how spherical trigonometry should act for instance.

Does one define the r as being the Aerial Coordinate, or any of its brethren, or as the Curvature of radius of the two-sphere about the central point. Both definitions of r will give the same answer for Schwarzschild Geometry, the interesting thing about interpreting r as radius of curvature is that it only works for spherically symmetric geometries.

I don't know what you mean by "the Gaussian Curvature of radius". You probably mean the Gaussian radius of curvature, which is something defined for 2D surfaces. The natural generalization of Gaussian curvature to 4D is the Ricci scalar - but that's identically zero for the Schwarzschild spacetime (because it's a solution to Einstein's equations in vacuum).

There are other curvature invariants that aren't zero for the Schwarzschild spacetime - like the Riemann tensor squared - and of course those depend in a simple way (as an inverse power, but including factors of the Schwarzschild radius in the numerator) on the Schwarzchild coordinate "r".

5. He challenges whether Astronomers have detected BH's or not because the test of a BH behaving as per current understanding is not whether there is a certain amount of energy-density in a given region (that does not test the predictions about BH's as per current consensus understanding), but tests of things like whether there is an event horizon or singularity (or singularities perhaps more realistically speaking?). Further, it is a complaint of his that both the event horizon and any singularities are unfalsifiable.

I agree with the complaint that just because a region has such and such a density of energy does not show that there is a BH there acting in the way that BH's are supposed to act according to current consensus. The problem is that energy density in some region is not the same as observing a singularity or event horizon (or perhaps other putative aspect of BH's). This is true.

I do agree that singularities are unfalsifiable, but I think that event horizons in principle constitute a falsifiable prediction. The aspect of light not being able to escape would probably be unfalsifiable, but it is asymptotically falsifiable to show that closer and closer to such a horizon things get smeared out as it were.

Direct imaging of the SgrA* event horizon (or, if you prefer, object resolution below that of the Schwarzschild radius) is likely to be announced (and appropriate papers published) some time in the next five years or so. Ditto that of the SMBH in the nucleus of M87. How? By VLBI in the (sub-)mm part of the microwave spectrum, with various baselines, up to (and perhaps beyond) the Earth's diameter.

Such imaging will certainly provide the possibility of good, almost direct, tests of various GR-based hypotheses concerning black holes (or at least their event horizons). Would you like some references?

The current astronomical evidence for the existence of black holes is a good deal more extensive, and wider ranging, than merely "a region has such and such a density of energy"! For example, in (some) HMXB (high mass x-ray binaries) matter seems to be transferred from the lower-mass object (usually an O or B star) to the higher-mass one ... and then disappear. Estimates of the mass of the more massive of the pair, in such HMBXs, put them all over the limit for gravitational bound neutron stars; for those HMXBs in which the matter does not seem to disappear, the estimated mass of the compact object in the pair is below this limit.

More direct evidence will come when the first unambiguous inspiral signal is observed by the likes of LIGO or VIRGO (probably within the next five years or so) ...

Hmm, I wonder though, do you have any references showing prior understanding by the physics community at large that the r in the Schw. Sol. is the Gaussian Curvature of radius (assuming you agree with that determination, but it is easy to check)?

As I understand the definition of Gaussian curvature, the Gaussian curvature of a Euclidean 2-sphere of radius r is 1/(r^2).

I am not a physicist, and can't speak for the physics community at large. It's easy to show that, for at least the past 40 years, the relativists have been aware of this relationship between Gaussian curvature and the Schwarzschild radial coordinate r. That relationship is implied by statements such as

Originally Posted by Hawking & Ellis

The coordinate r in this metric form is intrinsically defined by the requirement that 4πr2 is the area of these surfaces of transitivity.

In my previous post, I cited the equations given in MTW and in Wald, which make that relationship even more explicit than in Hawking&Ellis. I also quoted Wald's lucid explanation of the important distinction that must be made between the Schwarzschild radial coordinate r and the more intuitive notion of the distance between a 2-sphere and its center.

Originally Posted by tensordyne

What about the other points???

I think I agree with your four other points as stated. (In a couple of cases, you alluded to things on which we might disagree, but I'm not going to argue with things you haven't said.)

Erratum 2: In post #17, I twice wrote "Adams" when I meant "Abrams". Brain fart.

Yeah, the mathematicians have known since Gauss that in a spherically symmetric geometry that there will be a sub-manifold that is given by a sphere about the symmetry point that has radius of curvature that is constant, and thus can be used as a coordinate. I do not care to debate about the relevant merits of using the differing definitions of r.

I am glad I was careful enough not to step outside of any lines. It was hard at times. Maybe in a day or two now that this is covered I will cover why I think BH's are not a good prediction of physics as far as it is currently understood. Until then, its been enjoyable.

__________________I learned much from the Order of the Jesuits. Until now, there has never been anything more grandiose, on the earth, than the hierarchical organization of the Catholic church. I transferred much of this organization into my own party.

Direct imaging of the SgrA* event horizon (or, if you prefer, object resolution below that of the Schwarzschild radius) is likely to be announced (and appropriate papers published) some time in the next five years or so. Ditto that of the SMBH in the nucleus of M87. How? By VLBI in the (sub-)mm part of the microwave spectrum, with various baselines, up to (and perhaps beyond) the Earth's diameter.

Such imaging will certainly provide the possibility of good, almost direct, tests of various GR-based hypotheses concerning black holes (or at least their event horizons). Would you like some references?

The current astronomical evidence for the existence of black holes is a good deal more extensive, and wider ranging, than merely "a region has such and such a density of energy"! For example, in (some) HMXB (high mass x-ray binaries) matter seems to be transferred from the lower-mass object (usually an O or B star) to the higher-mass one ... and then disappear. Estimates of the mass of the more massive of the pair, in such HMBXs, put them all over the limit for gravitational bound neutron stars; for those HMXBs in which the matter does not seem to disappear, the estimated mass of the compact object in the pair is below this limit.

More direct evidence will come when the first unambiguous inspiral signal is observed by the likes of LIGO or VIRGO (probably within the next five years or so) ...

That is pretty interesting news (at least to me). I have been to LIGO (the facility in Washington State). Pretty neat. Disappearing mass huh. Crazy!

__________________I learned much from the Order of the Jesuits. Until now, there has never been anything more grandiose, on the earth, than the hierarchical organization of the Catholic church. I transferred much of this organization into my own party.

Oh yeah, I forgot, I would like some refs in answer to your previous question DRD.

__________________I learned much from the Order of the Jesuits. Until now, there has never been anything more grandiose, on the earth, than the hierarchical organization of the Catholic church. I transferred much of this organization into my own party.

If you know how to use ADS - click on "NASA ADS" under References and Citations at the bottom right - you will find a bunch of others on this topic (click on "References in the article" and "Referred Citations to the Article").

The object is SgrA*, as in a point source within the SgrA complex, which is the "A" (radio) source in the constellation of Sagittarius. "Sag" is used, sometimes, as an abbreviation for Sagittarius, in the name of (other) astronomical objects, e.g. the Sagittarius Dwarf Elliptical Galaxy (Sag DEG).

Possibly, but likely to tell us more about the gas cloud than the SMBH (supermassive black hole).

If the numbers are correct (I didn't check them), the event horizon is ~3 light-hours in radius, but the closest the cloud will get is ~36 light-hours. At that distance I doubt there'd be any strong-field effects which could be easily teased out of any observations we might make of it.

Oh, and that article uses both the correct name (SgrA*) and an incorrect one (SagA*)!

I don't think the Ricci scalar is a particularly natural generalization of Gaussian curvature. It's defined as the product of principal curvatures of a submanifold, and so in general not proportional to the Ricci scalar. Though this disagreement doesn't really affect the particular issue at hand.

Originally Posted by W.D.Clinger

It's easy to show that, for at least the past 40 years, the relativists have been aware of this relationship between Gaussian curvature and the Schwarzschild radial coordinate r.

Yes. Additionally, some books (e.g., Weinberg's) also treat Gaussian curvature as synonymous with Ricci scalar with a 1/2 factor.

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